Evaluation of Nørsett methods for integrating differential equations in time

Abstract

SUMMARYSome recently developed implicit time discretizations are discussed whose main application is to solving ordinary differential equations arising from finite element approximations to partial differential equations. Their theoretical properties, computer implementation and numerical behaviour, as observed in tests on simple examples, are compared with well-known discretizations such as the Crank-Nicholson method. INTRODUCI'IONIn the numerical solution of partial differential equations many techniques have been suggested for integrating forwards in time, ranging from simple finite difference methods such as the Crank-Nicolson method to sophisticated finite element methods in time. In the present paper time discretizations introduced by Nbrsett' are evaluated with emphasis on simplicity of implementation in computer programs. The new methods are shown to be competitive with Crank-Nicolson.Our sole concern is time discretization. Assuming that a partial differential equation has been discretized in its space dimensions to give a system of ordinary differential equations, it is this system which we consider. The fact that such a separation of variables is performed rules out of consideration some well known difference schemes as the Dufort-Frankel method which is not derived in this way.To be specific, consider the diffusion convection equation defined for 0 < x < 1 and positive time t with initial and boundary conditions:

Hierarchical higher-order dissipative methods for transient analysis

“…Taking into account for Equation (38), the solution of the above system leads to the recurrence scheme typical of N-methods (see Equation (20) or Equation (24)), as desired. Moreover, owing to the assumed basis functions, the approximate solution at the end of the current time step, y − i+1 , is actually given by Equation (26).…”

On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations

“…To circumvent this restriction, various Lo-stable algorithms based on extrapolation [3,8], Runge-Kutta [9], multiderivative [lo], and multistep [ l l ] formulas have been proposed. Methods corresponding to Lo-acceptable subdiagonal Pad6 approximations have also been considered (see [12] and [13]). For example, using the [0/1] Pad6 approximation to the exponential function in (1.4) results in the first-order Lo-stable implicit Euler method ( I -AtA)y(t + At) = y ( t ) .…”

“…Methods corresponding to Lo-acceptable subdiagonal Pad6 approximations have also been considered (see [12] and [13]). The expansion (1.9) is particularly useful in parallel computing environments because it can be used to apportion the work of solving the corresponding linear algebraic systems to processors operating concurrently, as suggested by Sweet [ 151.…”

High‐order hierarchical A‐ and L‐stable integration methods

“…The properties of an integration method depend on how the exponential matrix is approximated, while the load may be independently discretized in various ways, cf. Trujillo' or Smith et al 3 So as to concentrate on the former, we subsequently assume that the load is given by a constant vector f, in each interval.…”